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Theorem 2spthonot 30528
 Description: The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
Assertion
Ref Expression
2spthonot 2SPathOnOt SPathOn
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem 2spthonot
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 is2spthonot 30526 . . . 4 2SPathOnOt SPathOn
21adantr 465 . . 3 2SPathOnOt SPathOn
32oveqd 6212 . 2 2SPathOnOt SPathOn
4 simprl 755 . . 3
5 simprr 756 . . 3
6 xpexg 6612 . . . . . . 7
76anidms 645 . . . . . 6
8 xpexg 6612 . . . . . 6
97, 8mpancom 669 . . . . 5
109ad2antrr 725 . . . 4
11 rabexg 4545 . . . 4 SPathOn
1210, 11syl 16 . . 3 SPathOn
13 oveq12 6204 . . . . . . . 8 SPathOn SPathOn
1413breqd 4406 . . . . . . 7 SPathOn SPathOn
15 eqeq2 2467 . . . . . . . . 9
1615adantr 465 . . . . . . . 8
17 eqeq2 2467 . . . . . . . . 9
1817adantl 466 . . . . . . . 8
1916, 183anbi13d 1292 . . . . . . 7
2014, 193anbi13d 1292 . . . . . 6 SPathOn SPathOn
21202exbidv 1683 . . . . 5 SPathOn SPathOn
2221rabbidv 3064 . . . 4 SPathOn SPathOn
23 eqid 2452 . . . 4 SPathOn SPathOn
2422, 23ovmpt2ga 6325 . . 3 SPathOn SPathOn SPathOn
254, 5, 12, 24syl3anc 1219 . 2 SPathOn SPathOn
263, 25eqtrd 2493 1 2SPathOnOt SPathOn
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 965   wceq 1370  wex 1587   wcel 1758  crab 2800  cvv 3072   class class class wbr 4395   cxp 4941  cfv 5521  (class class class)co 6195   cmpt2 6197  c1st 6680  c2nd 6681  c1 9389  c2 10477  chash 12215   SPathOn cspthon 23559   2SPathOnOt c2pthonot 30519 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-2spthonot 30522 This theorem is referenced by:  el2spthonot  30532  el2spthonot0  30533  2spthonot3v  30538  2pthwlkonot  30547  2spotfi  30554  2spotdisj  30797
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